How to Graph Solutions to One-step and Two-step Linear Inequalities

Dive into the mysterious world of linear inequalities, where the solutions often lie in a vast stretch of the number line rather than a single, fixed point. This journey will explore one-step and two-step inequalities, inviting you to master the nuances of their graphical representations.

Step-by-Step Guide to Graphing Solutions to One-step and Two-step Linear Inequalities

Here is a step-by-step guide to graphing solutions to one-step and two-step linear inequalities:

Step 1: Understand the Terrain: Basic Inequalities Overview

• Less than ($$<$$): Think of a hungry alligator, always eager to chomp the smaller number.
• Greater than ($$>$$): The reverse. The alligator now wants the bigger number.
• Less than or equal to ($$≤$$): Here, the alligator doesn’t mind if it’s exactly equal or just a tad smaller.
• Greater than or equal to ($$≥$$): Big or just the same, both are good enough!

Step 2: Setting the Stage: Drawing a Number Line

• Take a ruler or a straightedge.
• Sketch a horizontal line, which will represent our number line.
• Evenly space and mark numbers on this line. For instance, from $$-10$$ to $$10$$.

Step 3: One-Step Inequalities: Baby Steps to Mastery

a) Isolate the Variable

i) If the inequality is $$x>5$$, then $$x$$ is already isolated.

ii) For an inequality like $$x+4<7$$, subtract $$4$$ from each side to get$$x<3$$.

b) Graph the Solution

i) For strict inequalities like $$x<3$$ or $$x>5$$:

• Find the number on the number line.
• Make an open circle on it, indicating the value is not included.
• Draw an arrow in the direction of the solution. For $$x<3$$, the arrow will point to the left.

ii) For inclusive inequalities like $$x≤3$$ or $$x≥5$$:

• Find the number.
• Fill in a solid circle, indicating that this value is part of the solution.
• Again, draw an arrow toward the solution side.

Step 4: Two-Step Inequalities: Double the Fun, Double the Challenge

a) Isolate the Variable

i) For an inequality like $$2x−3>7$$:

• Start by adding $$3$$ to each side: $$2x>10$$.
• Then, divide each side by $$2$$: $$x>5$$.

b) Graph the Solution

i) Much like one-step inequalities, locate the number on the line.

ii) Decide if you need an open or a solid circle based on the strictness or inclusiveness of the inequality.

iii) Draw the arrow pointing towards the solution’s direction.

Step 5: Celebrate the Complexity: Compound Inequalities

Sometimes, you might find a beast like $$3<x≤8$$. Here, $$x$$ is trapped between two numbers.

• Graph the two numbers. One will have an open circle $$(3)$$ and the other, a filled one $$(8)$$.
• Connect them with a line or segment to show $$x$$ can be any number in between.

Step 6: Conclusion: Embrace the Dance of Inequalities

Understanding and graphing linear inequalities is like a dance of logic and intuition. With time and practice, you’ll find the rhythm and soon be swirling and twirling through even the most intricate inequalities with grace and precision!

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